Coherent spin qubit shuttling through germanium quantum dots

Quantum links can interconnect qubit registers and are therefore essential in networked quantum computing. Semiconductor quantum dot qubits have seen significant progress in the high-fidelity operation of small qubit registers but establishing a compelling quantum link remains a challenge. Here, we show that a spin qubit can be shuttled through multiple quantum dots while preserving its quantum information. Remarkably, we achieve these results using hole spin qubits in germanium, despite the presence of strong spin-orbit interaction. In a minimal quantum dot chain, we accomplish the shuttling of spin basis states over effective lengths beyond 300 microns and demonstrate the coherent shuttling of superposition states over effective lengths corresponding to 9 microns, which we can extend to 49 microns by incorporating dynamical decoupling. These findings indicate qubit shuttling as an effective approach to route qubits within registers and to establish quantum links between registers.


INTRODUCTION
The envisioned approach for semiconductor spin qubits towards fault-tolerant quantum computation centers on the concept of quantum networks, where qubit registers are interconnected via quantum links [1].Significant progress has been made in controlling few-qubit registers [2,3].Recent efforts have led to demonstrations of high fidelity single-and two-qubit gates [4,5], quantum logic above one Kelvin [6][7][8] and operation of a 16 quantum dot array [9].However, scaling up to larger qubit numbers requires changes in the device architecture [10][11][12][13].
Inclusion of short-range and mid-range quantum links could be particularly effective to establish scalability, addressability, and qubit connectivity.The coherent shuttling of electron or hole spins is an appealing concept for the integration of such quantum links in spin qubit devices.Short-range coupling, implemented by shuttling a spin qubit through quantum dots in an array, can provide flexible qubit routing and local addressability [14,15].Moreover, it allows to increase connectivity beyond nearest-neighbour coupling and decrease the number of gates needed to execute algorithms.Midrange links, implemented by shuttling spins through a multitude of quantum dots, may entangle distant qubit registers for networked computing and allow for qubit operations at dedicated locations [14,[16][17][18].Furthermore, such quantum buses could provide space for the integration of on-chip control electronics [1], depending on their footprint.
Meanwhile, quantum dots defined in strained germanium (Ge/SiGe) heterostructures have emerged as a promising platform for hole spin qubits [30,31].The high quality of the platform allowed for rapid development of single spin qubits [32,33], singlet-triplet qubits [34][35][36], a four qubit processor [2], and a 4×4 quantum dot array with shared gate control [9].While the strong spin orbit interaction allows for fast and all-electrical control, the resulting anisotropic g-tensor [31,37] complicates the spin dynamics and may challenge the feasibility of a quantum bus.
Here, we demonstrate that spin qubits can be shuttled through quantum dots.These experiments are performed with two hole spin qubits in a 2×2 germanium quantum dot array.Importantly, we operate in a regime where we can implement single qubit logic and coherently transfer spin qubits to adjacent quantum dots.Furthermore, by performing experiments with precise voltage pulses and sub-nanosecond time resolution, we can mitigate finite qubit rotations induced by spin-orbit interactions.In these optimized sequences we find that the shuttling performance is limited by dephasing and can be extended through dynamical decoupling.

COHERENT SHUTTLING OF SINGLE HOLE SPIN QUBITS
Fig. 1.a shows a germanium 2×2 quantum dot array identical to the one used in the experiment [2].The chemical potentials and the tunnel couplings of the quantum dots are controlled with virtual gates (vP i , vB ij ), which consist of combinations of voltages on the plunger gates and the barrier gates.We operate the device with two spin qubits in quantum dots QD 1 and QD 2 and initialised The qubit is prepared in a superposition state using a π/2 pulse and is transferred to the empty quantum dot with a detuning pulse of varying amplitude, and then brought back to its initial position after an idle time.After applying another π/2 pulse we readout the spin state.e, i, Schematic illustrating the shuttling of a spin qubit between QD2 and QD3 (e) and between QD3 and QD4 (i).f, j, Charge stability diagrams of QD2-QD3 (f) and QD3-QD4 (j).To shuttle the qubit from one site to another, the virtual plunger gate voltages are varied along the detuning axis (white arrow), which crosses the interdot charge transition line.g, k, Probing of the resonance frequency along the detuning axis for the double quantum dot QD2-QD3 (g) and QD3-QD4 (k).The resonance frequencies of the spin in the different quantum dots are clearly visible, indicating the possibility to shuttle a hole while preserving its spin polarization.Nearby the charge transition, the resonance frequency cannot be resolved due to a combination of effects discussed in Supplementary Note 1. h, l, Coherent free evolution of a qubit during the shuttling between QD2-QD3 (h) and QD3-QD4 (l).Since the Larmor frequency varies along the detuning axes, the qubit initialized in a superposition state acquires a phase that varies with the idle time resulting in oscillations in the spin-up P ↑ probabilities.
the |↓↓⟩ state (see Methods).We use the qubit in QD 1 as an ancilla to readout the hole spin in QD 2 , using latched Pauli spin blockade [2,38,39].The other qubit starts in QD 2 and is shuttled to the other quantum dots by chang-ing the detuning energies (ϵ 23/34 ) between the quantum dots (Fig. 1.b, e and i).The detuning energies are varied by pulsing the plunger gate voltages as illustrated in Fig. 1.f and j.Additionally, we increase the tunnel cou-plings between QD 2 -QD 3 and QD 3 -QD 4 before shuttling to allow for adiabatic charge transfer.The g-tensor of hole spin qubits in germanium is sensitive to the local electric field.Therefore, the Larmor frequency (f L ) is different in each quantum dot [32][33][34].We exploit this effect to confirm the shuttling of a hole spin from one quantum dot to another.In Fig. 1.c.we show the experimental sequence used to measure the qubit resonance frequency, while changing the detuning to transfer the qubit.Fig 1.g (k) shows the experimental results for spin transfers from QD 2 to QD 3 (QD 3 to QD 4 ).Two regions can be clearly distinguished in between which f L varies by 110 (130) MHz.This obvious change in f L clearly shows that the hole is shuttled from QD 2 to QD 3 (QD 3 to QD 4 ) when applying a sufficiently large detuning pulse.To investigate whether such transfer is coherent, we probe the free evolution of qubits prepared in a superposition state after applying a detuning pulse (Fig. 1.d) [27].The resulting coherent oscillations are shown in Fig. 1.h (l).They are visible over the full range of voltages spanned by the experiment and arise from a phase accumulation during the idle time.Their frequency f osc is determined by the difference in resonance frequency between the starting and end point in detuning as shown in Supplementary Figure 1.The abrupt change in f osc marks the point where the voltage pulse is sufficiently large to transfer the qubit from QD 2 to QD 3 (QD 3 to QD 4 ).These results clearly demonstrate that single hole spin qubits can be coherently transferred.

THE EFFECT OF STRONG SPIN-ORBIT INTERACTION ON SPIN SHUTTLING
The strong spin-orbit interaction in our system has a significant impact on the spin dynamics during the shuttling.It appears when shuttling a qubit in a |↓⟩ state between QD 2 and QD 3 using fast detuning pulses with voltage ramps of 4 ns.Doing this generates coherent oscillations shown in Fig. 2.b that appear only when the qubit is in QD 3 .They result from the strong spin-orbit interaction and the use of an almost in-plane magnetic field [40].In this configuration, the direction of the spin quantization axis depends strongly on the local electric field [35,37,[41][42][43] and can change significantly between neighbouring quantum dots.Therefore, a qubit in a spin basis state in QD 2 becomes a superposition state in QD 3 when diabatically shuttled.Consequently, the spin precesses around the quantization axis of QD 3 until it is shuttled back (Fig. 2.a).This leads to qubit rotations and the aforementioned oscillations.
While these oscillations are clearly visible for voltage pulses with ramp times t ramp of few nanoseconds, they fade as the ramp times are increased, as shown in Fig. 2.c, and vanish for t ramp > 30 ns.The qubit is transferred adiabatically and can follow the change in quantization axis and therefore remains in the spin basis state in both quantum dots.From the visibility of the oscillations, we estimate that the quantization axis of QD 3 (QD 4 ) is tilted by at least 42 • (33 • ) compared to the quantization axis of QD 2 (QD 3 ).These values are corroborated by independent estimations made by fitting the evolution of f L along the detuning axes (see Supplementary Note 2).Fig. 2.d and Fig. 2.e display the magnetic field dependence of the oscillations generated by diabatic shuttling.Their frequencies f osc increase linearly with the field and match the Larmor frequencies f L measured for a spin in the target quantum dot.This is consistent with the explanation that the oscillations are due to the spin precessing around the quantization axis of the second quantum dot.

SHUTTLING PERFORMANCE
To quantify the performance of shuttling a spin qubit, we implement the experiments depicted in Fig. 3.a, e and f [15,27] and study how the state of a qubit evolves depending on the number of subsequent shuttling events.For hole spins in germanium, it is important to account for rotations induced by the spin-orbit interaction.This can be done by aiming to avoid unintended rotations, or by developing methods to correct them.An example of the first approach is transferring the spin qubits adiabatically.This implies using voltage pulses with ramps of tenths of nanoseconds, which are significant with respect to the dephasing time.However, this strongly limits the shuttling performance (see Supplementary Figure 5).Instead, we can mitigate rotations by carefully tuning the duration of the voltage pulses, such that the qubit performs an integer number of 2π rotations around the quantization axis of the respective quantum dot.This approach is demanding, as it involves careful optimization of the idle times in each quantum dot as well as the ramp times, as depicted in Fig. 3.b.However, it allows for fast shuttling, with ramp times of typically 4 ns and idle times of 1 ns, significantly reducing the dephasing experienced by the qubit during the shuttling.We employ this strategy in the rest of our experiments.
We first characterize the fidelity of shuttling spin basis states.We do this by preparing a qubit in a |↑⟩ or |↓⟩ state and transferring it multiple times between the quantum dots.Fig. 3.c and d display the spin-up fraction P ↑ measured as a function of the number of shuttling steps n.The probability of ending up in the initial state shows a clear exponential dependence on n.No oscillations of P ↑ with n are visible, confirming that the pulses have been successfully optimized to account for unwanted spin rotations.We find for the shuttling of basis states characteristic decay constants n * = 3000 shuttlings, corresponding to polarization transfer fidelities F = exp(−1/n * ) ≃ 99.97 %.This is similar to the fidelities reached in silicon devices [15,27], despite the anisotropic g-tensors due to the strong spin-orbit interaction in our platform.
We now focus on the performance of coherent shut-  tling.We prepare a superposition state via an EDSR (π/2) X pulse, shuttle the qubit, apply another π/2 pulse and measure the spin state.Importantly, one must account for ẑ-rotations experienced by the qubits during the experiments.Therefore, we vary the phase of the EDSR pulse ϕ for the second π/2 pulse.For each n, we then extract the amplitude A of the P ↑ oscillations that appear as function of ϕ [15,27].Fig. 3.g, h show the evolution of A as a function of n for shuttling between adjacent quantum dots.We fit the experimental results using A 0 exp (−(n/n * ) α ) and find characteristic decay constants n * 23 = 64 ± 1 and n * 34 = 77 ± 2. Remarkably, these numbers compare favourably to n * ≃ 50 measured in a SiMOS electron double quantum dot [27], where the spin-orbit coupling is weak.
The exponents, α 23 = 1.36±0.05and α 34 = 1.28±0.06,reveal that the decays are not exponential.This contrasts with observations in silicon [15,27], and suggests that the shuttling of hole spins in germanium is limited by other mechanisms.Two types of errors can be distinguished: those induced by the shuttling processes and errors due to the dephasing during free evolution.To investigate the effect of the latter, we modify the shuttling sequence and include a (π) X echoing pulse in the middle as displayed in Fig. 3.e.Fig. 3.g and h show the experimental results and it is clear that in germanium the coherent shuttling performance is improved significantly using an echo pulse: we can extend the shuttling by a factor of four to five, reaching a characteristic decay of more than 300 shuttles.Similarly, the use of CPMG sequences incorporating two decoupling (π) Y pulses (Fig. 3.f) allows further, though modest, improvements.These enhancements in the shuttling performance confirm that dephasing is limiting the shuttling performance contrary to observations in SiMOS [27].We speculate that the origin of the difference is two-fold.Firstly, due to the stronger spin-orbit interaction, the spin is more sensitive to charge noise, resulting in a shorter dephasing times [44].Secondly, the excellent control over the potential landscape in germanium allows minimizing the errors which are due to the shuttling itself.

SHUTTLING THROUGH QUANTUM DOTS
For distant qubit coupling, it is essential that a qubit can be coherently shuttled through a series of quantum dots.This is more challenging, as it requires control and optimization of a larger amount of parameters.We perform two types of experiments to probe the shuttling Detuning pulses are applied to the plunger gates to shuttle the hole from one quantum dot to another, back and forth, and finally the appropriate pulses are applied to prepare for readout.Moving the qubit from one quantum dot to another is counted as one shuttling event n = 1.Since the hole always needs to be shuttled back for readout, n is always an even number.The schematic shows an example for n = 6.b, Zoom-in on the detuning pulses used for the shuttling.To make an integer number of 2π rotation(s) around the quantization axis of the second quantum dot, all ramp and idle times in the pulse need to be optimized.c, d, Spin-up probabilities P ↑ measured after shuttling n times a qubit prepared in a spin basis state between QD2 and QD3 (c) and between QD3 and QD4 (d).The decay of P ↑ as a function of n is fitted to an exponential function P ↑ = P0 exp(−n/n * ) + Psat.e, Pulse sequence used for implementing a Hahn echo shuttling experiment.In the middle of the shuttling experiment, an echo pulse (π) X is applied in the quantum dot where the spin qubit was initially prepared.Example for n = 12.f, Pulse sequence for a CPMG shuttling experiment.Two (π) Y pulses are inserted between the shuttling pulses.
Example for n = 24.g, h, Performance of the shuttling of superposition state between QD2 and QD3 (g) and QD2 and QD3 (h) for different shuttling sequences.The decay of the coherent amplitude A of the superposition state are fitted by A0 exp (−(n/n * ) α ) where α is a fitting parameter.
through a quantum dot, labelled corner shuttling and triangular shuttling.Fig. 4.b shows a schematic of the corner shuttling, which consists of transferring a qubit from QD 2 to QD 3 to QD 4 and back along the same route.The triangular shuttling, depicted in Fig. 4.e, consists of shuttling the qubit from QD 2 to QD 3 to QD 4 , and then directly back to QD 2 , without passing through QD 3 (for the charge stability diagram QD 4 -QD 2 and a detailed description see Supplementary Note 4).
To probe the feasibility of shuttling through a quantum dot, we measure the free evolution of a coherent state while varying the detuning between the respective quantum dots.The results are shown in Fig 4 .a.We find a remarkably clear coherent evolution for hole spin transfer from QD 2 to QD 3 to QD 4 and to QD 2 .We observe one sharp change in the oscillation frequency for each transfer to the next quantum dot.We also note that after completing one round of the triangular shuttling, the phase evolution becomes constant, in agreement with a qubit returning to its original position.We thereby conclude that we can shuttle through quantum dots as desired.
We now focus on quantifying the performance of shuttling through quantum dots by repeated shuttling experiments.To allow comparisons with previous experiments, we define n as the number of shuttling steps between two quantum dots.Meaning that one cycle in the corner shuttling experiments results in n = 4, while a loop in triangular shuttling takes n = 3 steps.The results for   1.h and l for the corner and triangular shuttling processes.In these experiments, the amplitude of the detuning pulse is increased in steps, in order to shuttle a qubit from QD2 to QD3 and back (top panel), from QD2 to QD3 to QD4 and back (second panel).The measurement in the third panel is identical to the measurement in the second panel, but the final point in the charge stability diagram is stepped towards the charge degeneracy point between QD2 and QD4.In the bottom panel the qubit is shuttled in a triangular fashion: from QD2 to QD3 to QD4 to QD2.The ramp times for this experiment are chosen in such a way that the shuttling is adiabatic with respect to the changes in quantization axis.b, e, Schematic illustrating the shuttling of a spin qubit around the corner: from QD2 to QD3 to QD4 and back via QD3 (b) and in a triangular fashion: from QD2 to QD3 to QD4 and directly back to QD2 (e).The double arrow from QD4 to QD2 indicates that this pulse is made in two steps, in order for the spin to shuttle via the charge degeneracy point of QD4 -QD2 and avoid crossing charge transition lines.c, f, Performance for the corner shuttling (c) and the triangular shuttling (f) of a qubit prepared in the basis states.d, g, Performance for shuttling a qubit prepared in a superposition state for the corner shuttling (d) and the triangular shuttling (g) and for different shuttling sequences.Shuttling performance for different processes are summarized in Supplementary Table 1.
shuttling basis states are shown in Fig. 4.c and 4.f.We note that the spin polarization decays faster compared to the shuttling in double quantum dots, in particular for the triangular shuttling.The corresponding fidelities per shuttling step are F ≃ 99.96 % for the corner shuttling and F ≥ 99.63 % for the triangular shuttling.
For the corner shuttling, the faster decay of the basis states suggests a slight increase of the systematic error per shuttling.This may originate from the use of a more elaborated pulse sequence, which makes pulse optimization more challenging.Nonetheless, the characteristic decay constant n * remains above 2000 and corresponds to effective distances beyond 300 µm (taking a 140 nm quantum dot spacing).The fast decay for the triangular shuttling is likely originating from the diagonal shuttling step.The tunnel coupling between QD 2 and QD 4 is low and more challenging to control, due to the absence of a dedicated barrier gate.The low tunnel coupling demands slower ramp times (t ramp ≃ 36 ns) for the hole transfer.This increases the time spent close to the (1,1,0,0)-(1,0,0,1) charge degeneracy point where spin randomization induced by excitations to higher energy states is enhanced [45].
Remarkably, we find that the performance achieved for coherent corner shuttling (as shown in Fig. 4.d) are comparable to those of coherent shuttling between neighbouring quantum dots.This stems from the performance being limited by dephasing.However, the performance for the CPMG sequence appears inferior when compared to the single echo-pulse sequence.Since the shuttling sequence becomes more complex, we speculate that it is harder to exactly compensate for the change in quantization axes.Imperfect compensation may introduce transversal noise, which is not fully decoupled using the CPMG sequence.Moreover, close to the anticrossing, the spin is subject to high frequency noise [45], whose effect is not corrected and can be enhanced depending on the dynamical decoupling sequence.
The performance of the coherent triangular shuttling, displayed in Fig. 4.g, fall short compared to the corner shuttling.Yet, the number of shuttles reached remains limited by dephasing as shown by the large improvement of n * obtained using dynamical decoupling.The weaker performance are thus predominantly a consequence of the use of longer voltage ramps.A larger number of coherent shuttling steps may be achieved by increasing the diagonal tunnel coupling, which could be obtained by incorporating dedicated barrier gates.

CONCLUSION
We have demonstrated coherent spin qubit shuttling through quantum dots.While holes in germanium provide challenges due to an anisotropic g-tensor, we find that spin basis states can be shuttled n * = 2230 times and coherent states up to n * = 67 times and even up to n * = 350 times when using echo pulses.The small effective mass and high uniformity of strained germanium allow for a comparatively large quantum dot spacing of 140 nm.This results in effective length scales for shuttling basis states of l spin = 312 µm and for coherent shuttling of l coh = 9 µm.By including echo pulses we can extend the effective length scale to l coh = 49 µm.These results compare favourably to effective lengths obtained in silicon [15,[27][28][29].We note that using effective lengths to predict the performance of practical shuttling links requires caution, as the spin dynamics will dependent on the noise of the quantum dot chain.For example, if the noise is local, echo pulses may proof less effective.However, in that case, motional narrowing may facilitate the shuttling [22,25,29,46,47].Furthermore, operating at even lower magnetic fields and exploiting purified germanium will boost the coherence time and thereby the ability to coherently shuttle.
While we have focused on bucket-brigade-mode shuttling, our results also open the path to conveyor-mode shuttling in germanium, where qubits would be coherently displaced in propagating potential wells using shared gate electrodes.This complementary approach holds promise for making scalable mid-range quantum links and has recently been successfully investigated in silicon [29], though on limited length scales.However, for holes in germanium the small effective mass and absence of valley degeneracy will be beneficial in conveyor-mode shuttling.
Importantly, quantum links based on shuttling and spin qubits are realized using the same manufacturing techniques.Their integration in quantum circuits may provide a path toward networked quantum computing.

Materials and device fabrication
The device is fabricated on a strained Ge/SiGe heterostructure grown by chemical vapour deposition [30,48].From bottom to top the heterostructure is composed of a 1.6 µm thick relaxed Ge layer, a 1 µm step graded Si 1−x Ge x (x going from 1 to 0.8) layer, a 500 nm relaxed Si 0.2 Ge 0.8 layer, a strained 16 nm Ge quantum well, a 55 nm Si 0.2 Ge 0.8 spacer layer and a < 1 nm thick Si cap.Contacts to the quantum well are made by depositing 30 nm of aluminium on the heterostructure after etching of the oxidized Si cap.The contacts are isolated from the gate electrodes using a 10 nm aluminium oxide layer deposited by atomic layer deposition.The gates are defined by depositing Ti/Pd bilayers.They are separated from the each other and from the substrate by 7 nm of aluminium oxide.

Experimental procedure
To perform the experiments presented, we follow a systematic procedure composed of several steps.We start by preparing the system in a (1,1,1,1) charge state with the hole spins in QD 1 and QD 2 initialized in a |↓⟩ state, while the other spins are randomly initialized.Subsequently, QD 3 and QD 4 are depleted to bring the system in a (1,1,0,0) charge configuration.After that, the virtual barrier gate voltage vB 12 is increased to isolate the ancilla qubit in QD 1 .The tunnel couplings between QD 2 and QD 3 and, depending on the experiment, between QD 3 and QD 4 are then increased by lowering the corresponding barrier gate voltages on vB 23 and vB 34 .This concludes the system initialization.
Thereafter, the shuttling experiments are performed.Note that to probe the shuttling between QD 3 and QD 4 , the qubit is first transferred adiabatically (with respect to the change in quantization axis) from QD 2 to QD 3 .To determine the final spin state after the shuttlings, the qubit is transferred back adiabatically to QD 2 .Next, the system is brought back in the (1,1,1,1) charge state, the charge regime in which the readout is optimized.This is done by first increasing vB 34 and vB 34 , then decreasing vB 12 and finally reloading one hole in both QD 3 and QD 4 .We finally readout the spin state via latched Pauli spin blockade by transferring the qubit in QD 1 to QD 2 and integrating the signal from the charge sensor for 7 µs.Spin-up probabilities are determined by repeating each experiment a few thousand times (typically 3000).Details about the experimental setup can be found in ref. [2].

Achieving sub nanosecond resolution on the voltage pulses
The voltage pulses are defined as a sequence of ramps with high precision floating point time stamps and voltages.The desired gate voltage V (t) sequence is generated numerically, sampled at 1 GSa/s (maximum rate achievable with our setup) and then applied on the sample using arbitrary wave form generators (AWGs).To increase the resolution despite the finite sampling rate, we shift the ramps on the desired gate voltage sequence by fractions of nanoseconds.Shifting a ramp by τ results in a shift of the voltages by −τ dV (t)  dt .The AWGs outputting the voltage ramp have a higher order low-pass filter with a cut-off frequency of approximately 400 MHz that smoothens the output signal and effectively removes the effect of the time discretization.The time shift of a pulse is not affected by the filter as the time shift does not change the frequency spectrum of the pulse.Thus the voltage sequence effectively generated on the sample is only delayed by τ allowing to achieve a sub nanosecond resolution.The quantum dot where the shuttling experiment starts is taken as the reference point for the frequency.∆f is independently evaluated from measurements of the resonance frequency using an EDSR pulse (data displayed in Fig. 1.g and k) and from the frequency of the coherent oscillations that appear when a qubit is shuttled in a superposition state (data displayed in Fig. 1.h and l).Both sets of data points overlap in (a) and (b), confirming that coherent oscillations arise due to a change in Larmor frequency along the detuning axis.For the free evolution experiments, the shuttling between QD2 and QD3 (shown in (a)) is completely adiabatic (ramp times of 40 ns) while the shuttling between QD3 and QD4 (shown in (b)) is only partially adiabatic (ramp times of 4 ns).In the latter case, the frequency difference measured is barely affected by the limited adiabaticity as the visibility M of the oscillations induced by the change in quantization axis (M < 0.1 from Supplementary Figure 2) is sufficiently small compared to that of the oscillations arising from the phase evolution of the superposition state (V ≈ 0.5 when the hole is in QD4).Moreover, the Larmor frequency of both a spin in QD3 and in QD4 is very close to 1 GHz.The free evolution experiments were performed with 1 ns time precision, meaning that the oscillations due to the diabaticity of the shuttling only show up as an aliasing pattern and do not disturb the oscillations due to free evolution.the quantum dot where the shuttling experiment starts.We neglect relaxation which is irrelevant at the time scale of few nanoseconds [7] and thus assume that the norm of the vector state stays constant during the rotations.We find that: We use this expression to evaluate θ 23 (θ 34 ), the tilt angle between the quantization axes of QD 2 and QD 3 (QD 3 and QD 4 ).Supplementary Figure 2.b and c show the amplitudes M/2 of the oscillations induced by the change in quantization axis as function of the pulse ramp time t ramp .As discussed in the main text, the amplitudes drop rapidly to zero as t ramp increases, because the shuttling becomes more adiabatic with respect to the difference in quantization axis.For the evaluation of θ we use the amplitude M/2 = 0.14 (0.07) of the oscillations at the shortest t ramp = 2 ns.We remark that there is no clear saturation of M at the smallest ramp times, which suggests that the shuttling process is still not fully diabatic and that higher visibilities could be achieved by shuttling faster.Rabi oscillations for the driving of the qubit in QD 2 (QD 3 ) have a visibility of V = 0.61 (0.48) giving us θ 23 ≥ 42 • (θ 34 ≥ 33 • ).These large values for θ illustrate the strong influence of the local electric field on the direction of the quantization in germanium hole spin qubits operated with an in-plane external magnetic field.

B. Estimations based on fits with a four-level model
To get additional independent evaluation of the tilt angles, we can fit the evolution of the qubit resonance with a four-level model.To derive such a model, we consider a single hole in a germanium double quantum dot placed in an external magnetic field B. We assume that there is a finite tunnel coupling t c between the two quantum dots QD A and QD B and their quantization axis are tilted with respect to each other by an angle θ.This last assumption is sufficient to take into account all effects of the spin-orbit interaction, providing a suitable basis transformation and a renormalization of the tunneling terms.
The system can then be described in the basis {|A, ↑ A ⟩ , |A, ↓ A ⟩ , |B, ↑ A ⟩ , |B, ↓ A ⟩}, where 'A' or 'B' indicates the position of the hole in quantum dot QD A or QD B and ↑ A or ↓ A specifies its spin states in the frame of quantum dot A. Its Hamiltonian is then given by: where ϵ is the detuning energy of the double quantum dot system (taken as zero at the charge transition), µ B is the Bohr magneton and the g i are the g-factors in the different quantum dots, φ is the azimuthal angle between the two quantization axes.We note that this model is similar to that of a flopping-mode qubit [1].Diagonalizing the Hamiltonian, we obtain the qubit resonance frequency f L given by: The evolution of f L along the detuning axes can then be fitted to extract the tilt angles and the tunnel couplings between neighbouring quantum dots.For this purpose, we first express the detuning energies in terms of gate voltages as are the virtual gate lever arms measured nearby the QD 2 -QD 3 (QD 3 -QD 4 ) charge transition via photon-assisted tunnelling experiments [8] and where the γ ij = |∆vP i /∆vP j | are the slopes of the detuning axis.We then extract the evolution of f L as function of vP 3 (vP 4 ) from the data displayed in Supplementary Figure 3.a-b (Supplementary Figure 4.a-c) and fit it with eq.(3).Supplementary Figure 3.c-d display the evolution of f L along the ϵ 23 detuning axis which is fitted to the above model assuming a linear dependence of g with vP 3 .We observe that the model reproduces well the measured evolution.This allows to estimate an interdot tunnel coupling t c of 8.7 ± 0.3 GHz and a tilt angle θ 23 of = 51.8 ± 0.7 • .The latter is consistent with the lower bound found using the previous method.
Supplementary Figure 4.d-e display the evolution of f L along the ϵ 34 detuning axis.In this case, fitting the data does not allow to extract the tilt angle, even if we assume a quadratic dependence of the g-factor with the gate voltage.Indeed, for 0 • ≤ θ ≲ 40 • , the shape of f L curve is nearly solely determined by the tunnel coupling and the variation of the g-factor with vP 4 .Consequently, the data can be equally well fitted by models where θ 34 is fixed 0 • , 10 • , 20 • , 30 • or 40 • .This leads to large uncertainty on the value of θ 34 that prevents us to extract it.Nevertheless, the tunnel coupling between QD 3 and QD 4 can still be estimated from these fits and, for θ 34 fixed to 40 • (30 • ), we find t c = 15 ± 2 (t c = 12±2) GHz.
What does become clear, however, is that we cannot obtain proper fits of the data with model where θ 34 is fixed to values larger than 40 • .The underlying reason appears when plotting the expected evolution of f L in such model: for θ 34 ≳ 50 • , f L should display a minimum that we do not observe experimentally.This suggests that θ 34 is lower than 50 • .

Figure 1 .
Figure 1.Coherent shuttling of hole spin qubits in germanium double quantum dots.a, A false colored scanning electron microscope image of a similar device to the one used in this work.The quantum dots are formed under the plunger gates (light blue) and separated by barrier gates (dark blue) which control the tunnel couplings.A single hole transistor is defined by the yellow gates and is used as charge sensor.The scale bar corresponds to 100 nm.b, Schematic showing the principle of bucket brigade mode shuttling.The detuning energy ϵ 23/34 between the two quantum dots is progressively changed such that it becomes energetically favorable for the hole to tunnel from one quantum dot to another.c, Schematic of the pulses used for the shuttling experiments shown in (g) and (k), where the resonance frequency of the qubit is probed after the application of a detuning pulse using a 4 µs EDSR pulse.d, Schematic of the pulses used for coherent shuttling experiments of which the results are shown in (h) and (l).The qubit is prepared in a superposition state using a π/2 pulse and is transferred to the empty quantum dot with a detuning pulse of varying amplitude, and then brought back to its initial position after an idle time.After applying another π/2 pulse we readout the spin state.e, i, Schematic illustrating the shuttling of a spin qubit between QD2 and QD3 (e) and between QD3 and QD4 (i).f, j, Charge stability diagrams of QD2-QD3 (f) and QD3-QD4 (j).To shuttle the qubit from one site to another, the virtual plunger gate voltages are varied along the detuning axis (white arrow), which crosses the interdot charge transition line.g, k, Probing of the resonance frequency along the detuning axis for the double quantum dot QD2-QD3 (g) and QD3-QD4 (k).The resonance frequencies of the spin in the different quantum dots are clearly visible, indicating the possibility to shuttle a hole while preserving its spin polarization.Nearby the charge transition, the resonance frequency cannot be resolved due to a combination of effects discussed in Supplementary Note 1. h, l, Coherent free evolution of a qubit during the shuttling between QD2-QD3 (h) and QD3-QD4 (l).Since the Larmor frequency varies along the detuning axes, the qubit initialized in a superposition state acquires a phase that varies with the idle time resulting in oscillations in the spin-up P ↑ probabilities.

Figure 2 .
Figure 2. Rotations induced while shuttling by the difference in quantization axes.a, Schematic explaining the effect of the change in quantization axis direction that the qubit experiences during the shuttling process.The difference in quantization axis between quantum dots is caused by the strong spin-orbit interaction.b, Oscillations induced by the change in quantization axis while shuttling diabatically a qubit in a |↓⟩ state between QD2 and QD3.Ramp times of 4 ns are used for the detuning pulses.c, Oscillations due to the change in quantization axis at a fixed point in detuning, as function of the voltage pulse ramp time used to shuttle the spin.When the ramp time is long enough, typically above 30 ns, the spin is shuttled adiabatically and the oscillations vanish.d, Magnetic-field dependence of the oscillations induced by the difference in quantization axis.e, Frequency of the oscillations fosc induced by the change in quantization axis as a function of magnetic field for different shuttling processes.The oscillation frequency fosc for QD3 is extracted from measurements displayed in (d) (and similar experiments for the other quantum dot pairs) and is plotted with points.fosc scales linearly with the magnetic field.Comparing fosc with resonance frequencies measured using EDSR pulses (data points depicted with stars) reveals that fosc is given by the Larmor frequency of the quantum dot towards which the qubit is shuttled (black label).

QD 2 / 3 QD 3 Figure 3 .
Figure 3. Quantifying the performance for the shuttling in double quantum dots.a, Schematic of the pulse sequence used for quantifying the performance of shuttling basis states (blue) or a superposition state (grey).The spin qubit is prepared in the quantum dot where the shuttling experiment starts, by either applying an identity gate (shuttling a |↓⟩ state), a (π) X pulse (shuttling a |↑⟩ state) or (π/2) X pulse (shuttling a superposition state, also referred to as Ramsey shuttling experiments).Detuning pulses are applied to the plunger gates to shuttle the hole from one quantum dot to another, back and forth, and finally the appropriate pulses are applied to prepare for readout.Moving the qubit from one quantum dot to another is counted as one shuttling event n = 1.Since the hole always needs to be shuttled back for readout, n is always an even number.The schematic shows an example for n = 6.b, Zoom-in on the detuning pulses used for the shuttling.To make an integer number of 2π rotation(s) around the quantization axis of the second quantum dot, all ramp and idle times in the pulse need to be optimized.c, d, Spin-up probabilities P ↑ measured after shuttling n times a qubit prepared in a spin basis state between QD2 and QD3 (c) and between QD3 and QD4 (d).The decay of P ↑ as a function of n is fitted to an exponential function P ↑ = P0 exp(−n/n * ) + Psat.e, Pulse sequence used for implementing a Hahn echo shuttling experiment.In the middle of the shuttling experiment, an echo pulse (π) X is applied in the quantum dot where the spin qubit was initially prepared.Example for n = 12.f, Pulse sequence for a CPMG shuttling experiment.Two (π) Y pulses are inserted between the shuttling pulses.Example for n = 24.g, h, Performance of the shuttling of superposition state between QD2 and QD3 (g) and QD2 and QD3 (h) for different shuttling sequences.The decay of the coherent amplitude A of the superposition state are fitted by Ramsey, n* = 19(1) Hahn, n* = 78(3) QD 2 →QD 3 →QD 4 →QD 2 , n* = 2.7(3)e2 , n* = 3.8(4)e2 QD 2 →QD 3 →QD 4 →QD 2 Corner shuttling , n* = 2.28(7)e3 , n* = 2.23(8)e3 QD 2 →QD 3 →QD 4 →QD 3 →QD 2 Ramsey, n* = 67(2) Hahn, n* = 3.5(2)e2 CPMG, n* = 2.6Number of shuttles n QD 2 →QD 3 →QD 4 →QD 3 →QD 2 Idle time (ns) Number of shuttles n Number of shuttles n Number of shuttles n

Figure 4 .
Figure 4. Coherent shuttling through quantum dots.a, Results of free evolution experiments, similar to those displayed in Fig.1.hand l for the corner and triangular shuttling processes.In these experiments, the amplitude of the detuning pulse is increased in steps, in order to shuttle a qubit from QD2 to QD3 and back (top panel), from QD2 to QD3 to QD4 and back (second panel).The measurement in the third panel is identical to the measurement in the second panel, but the final point in the charge stability diagram is stepped towards the charge degeneracy point between QD2 and QD4.In the bottom panel the qubit is shuttled in a triangular fashion: from QD2 to QD3 to QD4 to QD2.The ramp times for this experiment are chosen in such a way that the shuttling is adiabatic with respect to the changes in quantization axis.b, e, Schematic illustrating the shuttling of a spin qubit around the corner: from QD2 to QD3 to QD4 and back via QD3 (b) and in a triangular fashion: from QD2 to QD3 to QD4 and directly back to QD2 (e).The double arrow from QD4 to QD2 indicates that this pulse is made in two steps, in order for the spin to shuttle via the charge degeneracy point of QD4 -QD2 and avoid crossing charge transition lines.c, f, Performance for the corner shuttling (c) and the triangular shuttling (f) of a qubit prepared in the basis states.d, g, Performance for shuttling a qubit prepared in a superposition state for the corner shuttling (d) and the triangular shuttling (g) and for different shuttling sequences.Shuttling performance for different processes are summarized in Supplementary Table1.

Supplementary Figure 3 .
Evaluation of the tilt angle between QD2 and QD3 quantization axes using a four-level model.a, Free evolution experiments for shuttling a qubit in superposition state between QD2 and QD3 back-and-forth.The superposition state is prepared in QD2.b, Zoom-in on the vicinity of the charge transition.The two data sets are identical to those displayed in Fig. 1.h.c, d, Resonance frequency extracted from the oscillations along the detuning axis in (a) and (b) and fit with the model of eq.(3).

Table 1 .
Shuttling processn * for |↓⟩ transfer n * for |↑⟩ transfer n * for |↓⟩+i|↑⟩ √ 2 transfer α for |↓⟩+i|↑⟩ √ Summary of shuttling performance.For the spin basis state shuttling experiments, the spin polarization decays with the number of shuttles n are fitted by P0 exp(−(n/n * )) + Psat.For the coherent shuttling experiments, the coherence decays are fitted by A0 exp (−(n/n * ) α ).n * represents the number of shuttles that can be achieved before the polarization the coherence or drops by 1/e.